How is the 130479.79 Value Calculated in this Retirement Calculator?

[brian110872]

On this chart:
http://www.banksite.com/calc/retire?with=30000&lngt=5&lngt=5&pay2=0.0&intr=7.5&outt=HTML+Tables
how did they calculate 130479.79? I know how they got the "Interest" and "End Bal" just not the 130479.79. Anyone have any suggestions?
Here is the original form:
http://www.banksite.com/calc/retire

Thank You,
Brian

 

[HallsofIvy]

In this example, the objective is to be able to withdraw $30,000 per year from savings. It assumes 7.5% interest (it's obviously very old!). The "end balance" is, of course,0 by definition.

Suppose you had X dollars in the bank, drawing 7.5% simple interest.
After 1 year, before you withdraw anything, you would have the original X plus interest, (0.075)(30000) or a total of (1+ 0.075)X= 1.075X. Now, you withdraw 30000. You have 1.075X- 30000 left.

For the second year, repeat that with initial amount 1.075X-30000. Before withdrawing anything, but including interest, you would have
1.075(1.075X- 30000)= (1.075)²X- (1.075)(30000). After withdrawing your 30000 you will have (1.075)²X- (1.075)(30000)- 30000 left.

For the third year, you are starting with that amount so, before withdrawal you would have 1.075((1.075)²X- 1.075(30000)- 30000)= 1.075³X- 1.075²300000- 1.075(30000). Now withdraw 30000 from that: 1.075³X- 1.075²300000- (1.075)30000- 30000= 1.075³X- 30000(1+ 1.075+ 1.075².

Do you see the pattern? After n years you will have 1.075ⁿX- (30000)(1+ 1.075+ ...+ 1.075^n-1. In particular, after 5 years you would have 1.075⁵X- (30000)(1+ 1.075+ 1.075²+ 1.075³+ 1.075⁴. Since in this example you are apparently only expecting to die 5 years after retirement, after 5 years, the "final balance" is to be 0. It's not that hard to calculate that 1+ 1.075+ 1.075²+ 1.075³+ 1.075⁴= 5.808 approximately. Solve the equation 1.075⁵X= 30000(5.808). I get slightly less than 130480. Try deducting the 30000 before adding the interest. That would be the same as replacing X by X-30000.

 

[brian110872]

Thanks for the reply HallsofIvy. For 1.075^5(X)= 30000(5.808), I'm getting X=121368. How am I miscalculating?

 

[Zurtex]

Thanks for the reply HallsofIvy. For 1.075^5(X)= 30000(5.808), I'm getting X=121368. How am I miscalculating?

You're not, you've solved for x almost perfectly, actually your missing a few decimal places there, can I assume you are using mathematica?

 

[brian110872]

You're not, you've solved for x almost perfectly, actually your missing a few decimal places there, can I assume you are using mathematica?

What is mathematica?

 

[brian110872]

HallsofIvy,
I'm not sure how you got 130480 from 1.0755X= 30000(5.808).

Thank You,
Brian